So you're OK with this, correct?

\(\displaystyle f(x) = \arctan \left(\frac{x^3}{4}\right) = \frac{x^3}{4} - \frac{x^9}{4^3 \cdot 3} + \frac{x^{15}}{4^5 \cdot 5} - \frac{x^{21}}{4^7 \cdot 7} + \cdots\)

Now, consider what happens when you take 9 derivatives of this expression.

The first term \(\displaystyle \frac{x^3}{4}\) will eventually become zero (beyond 3 derivatives).

The second term \(\displaystyle -\frac{x^9}{4^3 \cdot 3}\) will become a constant. What constant does it become?

The remaining terms will turn into the form \(\displaystyle C \cdot x^n\) where \(\displaystyle C\) is a constant and \(\displaystyle n\) is a positive integer. When you substitute \(\displaystyle x=0\) into a term of this form, does it make sense that these terms will all become zero?

Therefore, there is only one nonzero term, which is the second term.